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Gauss with Colum Pivot search

Posted on 2004-11-21
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Last Modified: 2016-02-10
Can someone explain me (with examples) the gaussian Algorithm with Column pivot search (I'm not english, so I'm not sure if it is the right name for it).

Thanks =)
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Question by:dkloeck
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LVL 31

Expert Comment

to explain Gauassian elimination with examples is a big job, and 30pts is not going to attract many people to this question.
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LVL 10

Author Comment

ok, i though it would be easy
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LVL 31

Accepted Solution

2y  +   z  = 1
2x  +  y  +  z   = 0
3x  + 3y +  z   = 2

0   2   1  |  1
2   1   1  |  0
3   3   1  |  2

find largest in col1 and swap row  to row1

3   3   1  |  2
2   1   1  |  0
0   2   1  |  1

divide row 1 by pivot value ie 3

1   1   1/3  |  2/3
2   1   1     |   0
0   2   1     |   1

eliminate in col 1

1    1   1/3  |  2/3
0   -1   1/3  |  -4/3
0    2   1     |   1

find largest in col2  diagonal and below and swap that row into row 2

1    1   1/3  |  2/3
0    2   1     |  1
0   -1   1/3  |  -4/3

divide  by pivot value, ie 2

1    1   1/3  |  2/3
0    1   1/2  |  1/2
0   -1   1/3  |  -4/3

eliminate in col 2

1    1   1/3  |  2/3
0    1   1/2  |  1/2
0    0   5/6  |  -5/6

ie

x  +  y  + 1/3z   =  2/3
y +  1/2z   =  1/2
5/6z   = -5/6

now back substitute

5/6z   = 5/6   =>  z=-1

y +  1/2z   = 1/2    =>    y = 1/2- 1/2z =1

x  +  y  + 1/3z   =  2/3  =>x    =  2/3 - y  -1/3z  = 1-1/3-2/3 = 0

x=0  y=1  z=-1

(substituting in to orignal equations for a check shows this is correct)

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LVL 31

Expert Comment

shld read            5/6z   =- 5/6   =>  z=-1
0

LVL 10

Author Comment

it would be enough if someone gives me an example on a 3x3 System, for example:

(2  1 -2) (x)   (10)
(3  2  2)·(y)= (1)
(5  4  3) (z)   (4)

I think this one is positiv definit, so it can be made.
Please make it step by step with some explanation

Thanks
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LVL 10

Author Comment

didnt see that before ^_^
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LVL 31

Expert Comment

Thx for the points. I will note that there are 2 similar algorithms , one called (1) Gaussian elimation which forms a triangular matrix and backsubstitutes and (2) Gauss-Jordan that forms a unit matrix and reads the values off. (1) is faster for the computer (2) is easier by hand (ie for humans)
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